系統識別號 | U0002-0307201317280800 |
---|---|
DOI | 10.6846/TKU.2013.00107 |
論文名稱(中文) | Reeb定理上的Morse理論及其推廣 |
論文名稱(英文) | Morse Theory to Reeb's Theorem and its Generalization |
第三語言論文名稱 | |
校院名稱 | 淡江大學 |
系所名稱(中文) | 數學學系碩士班 |
系所名稱(英文) | Department of Mathematics |
外國學位學校名稱 | |
外國學位學院名稱 | |
外國學位研究所名稱 | |
學年度 | 101 |
學期 | 2 |
出版年 | 102 |
研究生(中文) | 李容瑄 |
研究生(英文) | Jung-Hsuan Lee |
學號 | 600190069 |
學位類別 | 碩士 |
語言別 | 英文 |
第二語言別 | |
口試日期 | 2013-06-26 |
論文頁數 | 27頁 |
口試委員 |
指導教授
-
余成義(cherngyi@mail.tku.edu.tw)
委員 - 謝忠村(ctshieh@mail.tku.edu.tw) 委員 - 林吉田(ctlin@pu.edu.tw) |
關鍵字(中) |
Morse引理 Morse定理 Reeb定理 |
關鍵字(英) |
Morse lemma Morse Theorem Reeb's Therem |
第三語言關鍵字 | |
學科別分類 | |
中文摘要 |
本文先介紹可微流型之定理及各種基本性質,也初略介紹具Riemann測度之可微流型。 本文主要內容在使用Morse引理及Morse定理,重新述明Reeb定理之證明。這裡我們使用Surgery引理,處理邊界問題。 |
英文摘要 |
In this thesis, we want to use Morse Theorem to prove Reeb's Theorem. Before showing the proof of these theorems, we need to review some basic properties of a differentiable manifold M with a differentiable structure. In general, if we define some functions from M (or its subspace) to real value, the difference between manifold and coordinate space should be considered. Every point we choose must send to a coordinate subspace first. So defining a coordinate system is helpful to deal with any functions on manifold M. The main result we review is to prove Reeb's Theorem using Morse Lemma and Morse Theorem. Here we use a surgery lemma to prove disjoint union of two spaces, matched along their common boundary. We also show how to construct a homotopy equivalence between manifold M and a n-sphere, for all dimension n is larger or equal to 1. |
第三語言摘要 | |
論文目次 |
Contents 0 Introduction 1 1 Di erentiable manifold 3 2 Non-degenerate Critical Point And Hessian 7 3 Smooth Vector Field on Manifold 9 4 Homotopy 13 5 Morse Lemma 15 6 Main Theorem 18 7 State of Reeb's Theorem(construct a homeomorphism between M and S^n) 23 8 Conclusion and Remarks 26 |
參考文獻 |
[1] William M. Boothby. An Introduction to Differentiable Manifolds and Rieman- nian Geometry. ACADEMIC PRESS,INC, Orlando,Florida, second edition, 1986. [2] J.Milnor. On manifolds homeomorphic to the 7-sphere. Annals of Mathematics, 64:399{405, 1956. [3] James R. Munkres. Elements of Algebraic Topology. rst edition. |
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